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Optical Engineering

Departures From The Optical Bloch Equations

A. A. Villaeys

Published Jun 1, 1983• 0 citations
Publisher

Abstract

It is the aim of the present work to show some departures from the well-known optical Bloch equations usually obtained when the damping of a two-level system driven by a Glauber coherent mode of the radiation field is treated in the Markovian approximation. In particular, we show that for an arbitrary intensity we have a set of 4 n optical Bloch equations which can be reduced to the usual optical Bloch equations when the population of the mode becomes important. A criterion for the quasi-classical evolution can be deduced. Obviously, in both limits of an arbitrarily weak or strong excitation intensity the Mollow's results are obtained. In addition, we do not need here the introduction of a particular form like the plane wave used by Mollow for the corresponding classical field--as the problem is solved explicitly.

DOI
10.1117/12.7973106

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It is the aim of the present work to show some departures from the well-known optical Bloch equations usually obtained when the damping of a two-level system driven by a Glauber coherent mode of the radiation field is treated in the Markovian approximation. In particular, we show that for an arbitrary intensity we have a set of 4 n optical Bloch equations which can be reduced to the usual optical Bloch equations when the population of the mode becomes important. A criterion for the quasi-classical evolution can be deduced. Obviously, in both limits of an arbitrarily weak or strong excitation intensity the Mollow's results are obtained. In addition, we do not need here the introduction of a particular form like the plane wave used by Mollow for the corresponding classical field--as the problem is solved explicitly.
The master equation for a two-level atom driven by a strong classical field and damped into a tailored reservoir with nonflat density of modes is derived under the Born-Markov approximation. To derive the master equation, the dressing transformation on the atomic operators is performed first, and, next, the dressed operators are coupled to the reservoir and the corresponding damping rates are calculated. The effects of a strong field and/or structured reservoir are seen as nonstandard terms in the master equation, some of which are reminiscent of terms known for squeezed vacuum reservoirs. The master equation leads to the generalized optical Bloch equations that can easily be solved for the steady state and, together with the quantum regression theorem, allow for analytical expressions for the fluorescence, as well as absorption spectra.
A set of coupled non-Markovian equations of motion is derived for the Green's function of a driven two-level atom. These equations can be expressed in terms of the density-matrix elements only in the Markovian limit, in which case the set reduces to optical Bloch equations. The non-Markovian effect arises from correlations between successive photon emissions, an effect which is neglected in the existing equations of motion. The resulting correction is of the order of ${\ensuremath{\gamma}}_{0}$/${\ensuremath{\Omega}}_{0}$ in the strong-field region (${\ensuremath{\gamma}}_{0}$${\ensuremath{\Omega}}_{0}$) and it vanishes in the weak-field region, where ${\ensuremath{\gamma}}_{0}$ is the Einstein A coefficient and ${\ensuremath{\Omega}}_{0}$ is the Rabi frequency.
We introduce a non-Hermitian Schrödinger-type approximation of optical Bloch equations for two-level systems. This approximation provides a complete and accurate description of the coherence and decoherence dynamics in both weak and strong laser fields at the cost of losing accuracy in the description of populations. In this approach, it is sufficient to propagate the wave function of the quantum system instead of the density matrix, providing that relaxation and dephasing are taken into account via automatically adjusted time-dependent gain and decay rates. The developed formalism is applied to the problem of scattering and absorption of electromagnetic radiation by a thin layer comprised of interacting two-level emitters.
The optical Bloch equations, which give the time evolution of the elements of the density matrix of an atomic system subject to radiation, are generalized so that they can be applied when transitions between pairs of states can proceed by more than one stimulated route. The case considered is that for which the time scale of interest in the problem is long compared with that set by the differences in detuning of the radiation fields stimulating via the different routes. It is shown that the Bloch equations then reduce to the standard form of linear differential equations with constant coefficients. The theory is applied to a two-state system driven by two lasers with different intensities and frequencies and to a three-state Λ-system with one laser driving one transition and two driving the second. It is also shown that the theory reproduces well the observed response of a cold 40Ca+ ion when subject to a single laser frequency driving the 4S1/2–4P1/2 transition and a laser with two strong sidebands driving 3D3/2–4P1/2.
The generalized master equation for a two-level atom driven by a strong classical field and damped into a ‘tailored’ reservoir with a non-flat density of modes is derived under the Born-Markov approximation. To derive the master equation the dressing transformation on the atomic operators is performed first and next the dressed operators are coupled to the reservoir and the corresponding damping rates are calculated. The modifications introduced by a strong field and/or by the reservoir with the non-flat density of modes lead to non-standard terms in the master equation, some of which are reminiscent of terms known for squeezed vacuum reservoirs. The optical Bloch equations based on this generalized master equation are obtained and solved for the steady state. The solutions are discussed from the point of view of both bare and dressed atoms. Analytical formulas for the fluorescence and probe absorption spectra are obtained and illustrated graphically.
The isolation usually encountered in optics between the part of a field that is of interest and its source motivates the consideration of a radiation mode weakly coupled to a quantum-mechanical source. After the introduction of some refinements into the quantum mechanics of a damped radiation mode, the field is expressed as the sum of two parts, one due to the source (the source field) and the other due to the loss mechanism (the "vacuum" field). The characteristic function for the field is calculated up to second order in perturbation theory. This function is then compared with the characteristic function for the field in the presence of a classical source. A method is exhibited by which a classical source can be found such that the two characteristic functions are identical when averaged over a half cycle. In particular, the two sources yield the same expectation values for the instantaneous amplitude and energy of the field. The description of the equivalent classical source must be given in statistical terms, in general, and requires only a knowledge of 〈S(0)(t)〉〈{S}^{(0)}(t)〉〈S(0)(t)〉 and 〈S(0)(t1)S(0)(t2)〉〈{S}^{(0)}({t}_{1}){S}^{(0)}({t}_{2})〉〈S(0)(t1​)S(0)(t2​)〉, where S(0){S}^{(0)}S(0) is the dipole-moment operator of the quantum-mechanical source unperturbed by the mode under consideration (but otherwise arbitrarily complex, with the possibility of strong coupling to other modes). The theory is illustrated by a consideration of several simple sources---a two-level system, a harmonic oscillator, and a blackbody---for which equivalent classical sources are found. The two-time correlation functions for the field obtained with the two types of sources are compared and are shown to be the same up to first order in $\ensuremath{\xi}\ensuremath{\tau}$, where $\ensuremath{\tau}$ is the difference between the two times and $\ensuremath{\xi}$ is the inverse of the field relaxation time; the physical meaning of the second-order difference in the correlation functions is discussed. A limiting process, in which both the coupling to the source and the damping become small, is suggested as a method of adapting the results to free fields, but it is pointed out that for discussion of a single mode, a free field is physically less satisfactory than a damped field. It is concluded that, within a reasonable approximation scheme, the source field may be described classically (the "vacuum" field furnishing all the necessary quantum-mechanical properties of the total field).